Generalized LM-inverse of a matrix augmented by a column vector

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Abstract

This paper presents a direct proof of the recursive formulae for the generalized LM- inverse of a matrix augmented by a column vector. The recursive relations are proved by direct verification of the four conditions of the generalized LM-inverse. Several auxiliary results pertaining to generalized inverses are also provided.

Introduction

Let us begin by considering a set of linear equationsBx=b,where B is an m by n matrix, b is an m-vector, and x is an n-vector.

The generalized LM-inverse of the matrix B is the matrix such that the solutionx=BLM+bminimizes bothG=L1/2(Bx-b)2=Bx-bL2andH=M1/2x2=xM2,where L is an m by m symmetric positive definite matrix and M is an n by n symmetric positive definite matrix.

Below are the four conditions for the generalized LM-inverse [1].(i)BBLM+B=B,(ii)BLM+BBLM+=BLM+,(iii)LBBLM+is symmetric,(iv)MBLM+Bis symmetric.We note that the generalized LM-inverse is the more general kind of the Moore–Penrose inverse. The concept of Moore–Penrose (MP) inverses was first introduced by Moore [2] in 1920 and later independently by Penrose [3] in 1955. In 1960, Greville [4] gave the first formulae for recursively determining the Moore–Penrose inverse of a matrix. His algorithm provides an update of the MP inverse of a matrix whenever new information becomes available. As a result, the recursive formulae have found extensive use in many areas of applications. Among them are statistical inference [5], filtering theory, estimation theory [6], system identification [7], optimization and control, and most recently analytical dynamics [8], [9]. In 1997 Udwadia and Kalaba [10] provided an alternative and simple constructive proof of Greville’s formulae, and later [11], [12] developed recursive relations for different types of generalized inverses of a matrix including the least-squares generalized inverse, the minimum-norm generalized inverse, and the Moore–Penrose (MP) inverse of a matrix.

Recently, the recursive formulae for the generalized M-inverse [13], [14] and for the generalized LM-inverse were obtained. Those for the generalized LM-inverse were proved constructively [15]. In this paper, we provide a much simpler and alternative proof for the recursive formulae of the generalized LM-inverse, BLM+, of any given matrix, B, partitioned as B=[A|a], where A is an m by n  1 matrix and a is a column vector of m components. We show that the four conditions of the generalized LM-inverse of the recursive formulae are satisfied. Besides its inherent simplicity, our proof requires several subsidiary properties of the generalized LM-inverse of a matrix, many of which appear to be hereto unknown; they are presented in the Appendix. More general than the generalized M-inverse, the generalized LM-inverse finds use in an even wider range of application areas than the Moore–Penrose inverse – areas ranging from system theory, statistics, filtering, control theory, and optimization, to signal processing and mechanics.

Section snippets

Result

For any given matrixB=[A|a]its generalized LM-inverse formulae are given byBLM+=[A|a]LM+=ALM-+-ALM-+adL+-pdL+dL+,whend=(I-AALM-+)a0,=ALM-+-ALM-+ah-phh,whend=(I-AALM-+)a=0,where A is an m by (n  1) matrix, a is a column vector of m components, dL+=dTL/(dTLd), h=1βqTMU, β = qTMq, U=ALM-+0m, q=v+p-1, p=(I-ALM-+A)M--1m˜, and v=AM-+a. Note that L is a symmetric positive definite m by m matrix, andM=M-m˜Tm˜m¯,where M is a symmetric positive definite n by n matrix, M is the symmetric positive definite (n

Conclusions

The recursive formulae for obtaining the generalized LM-inverse of any general matrix augmented by a column vector were first given in Ref. 15. There they were derived in a constructive manner. We herein provide an alternative proof of their formulae by directly verifying that the four conditions of the generalized LM-inverse are satisfied, thereby confirming the validity of the formulae, and providing several new auxiliary results related to these generalized inverses.

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